{
  "abstract": "In this work, we establish a criterion for classifying elliptic curves that have no solutions in the set of positive integers. In particular, we prove that if $E: y^2 = Ax^3 + Bx^2 + Cx + D$ is an elliptic curve where $A$, $B$, $C$, and $D$ are coefficients of the curve, and $p$ is a prime number where $p \\equiv 1 \\pmod{4}$, and $(x,y)$ is a positive integer point, then if $$ -3A + C \\equiv -1 \\pmod{4}, $$ $$ A - B + C \\equiv -1 \\pmod{4}, $$ and $$ p^2 = A - B + C - B, $$ then there are no positive solutions for the curve $E$, where $E(\\mathbb{Z}^{+}) = \\varnothing$. Furthermore, we generalize this criterion by incorporating techniques from linear algebra. Specifically, we introduce a matrix construction to establish the nonexistence of positive integer solutions for a family of elliptic curves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22344",
  "author_total": 1,
  "authors": [
    {
      "name": "Shazali Abdalla Fadul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rbzvh7wey5kk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22344",
  "fulltext_url": null,
  "id": "1rbzvh7wey5kk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22344",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214021526",
  "pdf_url": "https://arxiv.org/pdf/2609.22344",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rbzvh7wey5kk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22344",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22344"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Criterion for classifying Elliptic curve that are unsolvable in the set integers",
  "title_text": "A Criterion for classifying Elliptic curve that are unsolvable in the set integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/1rbzvh7wey5kk",
  "venue": "arXiv",
  "versions": []
}